This refers to the page https://en.wikipedia.org/wiki/Rough_path where it mentions about rough path and regularity structure as explains in the page:https://en.wikipedia.org/wiki/Regularity_structure It is understandable that regularity structure uses some concepts of rough path theory. I feel that rough path is like a one kind of regularity structure and there are more other structures. Is this correct? Can anyone write a bit to show the clear connection between the rough path and regularity structure?
2026-03-25 08:07:11.1774426031
What is the connection between regularity structure and rough path theory?
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I am answering it only to accept the request from @Marine Galantin. My answer may be completely wrong or too simple and obvious and useless, if so my sincere apology.
Rough path theory and regularity structure has one common theme i.e 'model'. A model which can replace the derivatives and coefficients of Taylor series. Taylor series or Taylor polynomial is fine for continously differentiable function but if the function is continuous but non-differentiable (like brownian motion) what can we do. To fix this problem we need a model which has the properties of Taylor series. To do so, few new "objects" were needed to be created. The "signature" of the rough path theory is a model of the regularity structure. The justfication and bound of those new "objects" are developed and called: a) Rough path theory for function which are solution to SDE and b) Regularity structure for function solution to SPDE.